Characterizing Shape Theories by Kan Extensions

Topology and its Applications 120 (2002) 355–363

Characterizing shape theories by Kan extensions

L. Stramaccia

Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli, I-06123 Perugia, Italy Received 18 February 2000; received in revised form 16 January 2001

Abstract Every pair (C, K) of categories, where K is a proreﬂective subcategory of C, generates a shape theory. As a main result in this paper we give a characterization of such pairs, showing that these are exactly those having the property that every functor F : K → A has a Kan extension Ran F : C → A, which is preserved by all functors commuting with inverse limits.  2002 Elsevier Science B.V. All rights reserved. AMS classiﬁcation: 55P55; 54C56; 18A40; 18B30

Keywords: Pre-shape theory; Proreﬂector; Cechˇ extension; Kan extension

1. Introduction

Given a pair of categories (C, K), there exist essentially two ways for extending a functor deﬁned on the subcategory K to all of C, namely the Cechˇ and the Kan procedure. The Cechˇ method works for C a category of topological spaces and K a suitable subcategory of polyhedra (see [13,5]), while the Kan method (see [17]) is of a very general nature. Dold [7] pointed out that the Cechˇ process applies to a great variety of situations and that it coincides with the Kan process for homotopy functors. In [5] Deleanu and Hilton were concerned with the problem of when the Cechˇ extension of a general cohomology theory is again a cohomology theory and realized that this happens exactly when such an extension is actually a Kan extension. In particular they proved their result for a so called Cechˇ extension category (C, K) [13], where C is a category of paracompact spaces. One more insight in the theory of Cechˇ extensions is given by the fact that it is on such a procedure that the theory of shape is based [16]. The aim of this note is to provide a general framework to describe the Cechˇ and Kan procedures for extending functors and to give, in the very general setting, the converse of

E-mail address: [email protected] (L. Stramaccia).

0166-8641/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0166-8641(01)00086-4 356 L. Stramaccia / Topology and its Applications 120 (2002) 355–363 the result by Deleanu and Hilton quoted above. In Propositions 2.3 and 2.5 we show how the Cechˇ process may be generalized to the case of an (embedding) functor E : K → C having a proadjoint, in such a way that every functor F : K → A, with A having inverse limits, has an extension F:ˇ C → A andthenprovethatFˇ is actually the right Kan extension of F along E. This situation was already considered, from different points of view, also in [1,4,6,8]. We then go on proving a formal converse of such result: if an embedding functor E : K → C produces Kan extensions for each F : K → A, with A as above, then these extensions have indeed to be of Cechˇ type (Theorem 3.1). In so doing we also obtain a characterization of what we call here pre-shape theories, in terms of extension of functors.

2. Extension properties

An inverse system in a category M is a contravariant functor X : I → M,where I = (I, ) is a directed set. In the following we shall write X = (Xi,xii ,I),whereXi = X(i),foreveryi ∈ I,andxii = X(i i ) : Xi → Xi . Clearly, the bonding morphisms xii of the inverse system satisfy the relations: = ∈ (1) xii 1Xi ,foreveryi I,and (2) xii · xii = xii ,foreveryi i i in I. The class of all inverse systems in M can be organized in a category denoted ProM. For all details concerning its deﬁnition we refer to [16]. We only recall that a morphism p : Y → X,whereY ∈ M and X is as above, is given by a family

p ={pi : Y → Xi | i ∈ I} of morphisms of M, such that xii · pi = pi ,foreveryi i . There is an embedding eM : M → Pro M and, moreover, Pro M has inverse limits and it can be considered as a completion of M with respect to them. If A is any category having inverse limits, then every functor T : M → A has an extension T∗ :ProM → A, according to the following construction: • let Pro T : ProM → Pro A be the functor deﬁned on X = (Xi,xii ,I),byProT(X) = (T(Xi), T(xii ), I), • let lim : Pro A → A be the inverse limit functor, then T∗ = lim · ProT. Let Cat denote the metacategory of all categories and let Cat∗ be its subcategory of all categories having inverse limits. The construction above then shows the metafunctor ∗ Pro : Cat → Cat , M → ProM, as a left adjoint to the inclusion Cat∗ ⊂ Cat, hence one has a reﬂective situation as pictured in the following commutative diagram e M M Pro M

T T∗ A L. Stramaccia / Topology and its Applications 120 (2002) 355–363 357 for every T : M → A, A ∈ Cat∗.HereT∗ is uniquely determined only up to isomorphisms, unless a prescribed choice for inverse limits in A is given.

Deﬁnition 2.1. Let K be a full subcategory of C.AK-expansion of an object X ∈ C consists of an inverse system X = (Xi ,xii ,I) ∈ Pro K and a morphism p : X → X in ProC, having the following universal property: for every morphism f : X → K, K ∈ ProK, there is a unique morphism g : X → K in Pro K, which renders the following diagram commutative: p X X

f g K

Let us note that, when a K-expansion X exists for an object X ∈ C, then it is uniquely determined up to isomorphisms. The subcategory K is said to be proreﬂective in C whenever every object X ∈ C admits a K-expansion p : X → X.

Let us recall that, if K is proreflective in C, then the embedding E : K → C admits a proadjoint [4,10] P : C → Pro K, which is defined as follows: P(X) = X,foreveryX ∈ C, while for f : X → Y and q : Y → Y a K-expansion, P(f)= f is the unique morphism such that f · p = q · f . For every other choice of the K-expansions, we obtain a functor which is naturally isomorphic to P. If P is proadjoint to E, then P∗ :ProC → ProK is actually a reflector [19]. Conversely, assuming that Pro K is reflective in Pro C via R : Pro C → Pro K, then R · eC : C → Pro K is the proadjoint to E. Shape Theory was founded by Borsuk [2] in his study of homotopy properties of compacta. Soon after, Holsztynski´ [12] observed that shape could be formulated in an abstract categorical setting. Mardešic´ in [15] extended the notion of shape to arbitrary topological spaces, also recognizing the essential categorical features of the theory. In fact, he defined the shape category Sh, for the inclusion HPol → HTop and the shape functor S:HTop → Sh, characterizing the pair (Sh, S) by means of an appropriate universal property. Later on, Le Van [14] introduced the notion of shape for full embeddings K ⊂ C of abstract categories and Deleanu and Hilton [6] further extended and studied a categorical notion of shape for arbitrary functors E : K → C. See also the work of Frei [8]. In [4], Cordier and Porter have given a definition for what has to be a shape theory in the very general setting, starting from an arbitrary functor E : K → C, and discuss the Holsztynski´ shape theory (ShE, SE) as a major example. Here ShE is the category having the same objects as C, while a morphism t ∈ ShE(X, Y ) is a natural transformation t:C(Y, E−) → C(X, E−).SE : C → ShE is the identity on objects and assigns to every ∗ f : X → Y the natural transformation SE(f ) = f , induced by f . In case E is the embedding of a full proreflective subcategory K of C, then [4, p. 57] one has ShE(X, Y ) = ProK(X,Y ),wherep : X → X and q : Y → Y are selected K-expansions. It follows that 358 L. Stramaccia / Topology and its Applications 120 (2002) 355–363 any pair (C, K),whereK is a full proreflective subcategory of C, generates a (Holsztynski)´ shape theory in the previous sense. Hence we may consider any such a pair (C, K) as the germ for a shape theory and call it a pre-shape theory. This is consistent with our point of view that a shape theory should be intended as a theory of approximation of the objects of a category C by means of inverse systems taken in a subcategory K, whose objects are considered to be good to some extent. This happens in the classical Shape Theory [16] which is based on the proreflectivity of the homotopy category of polyhedra HPol in the homotopy category of all topological spaces HTop.

Example 2.2. (1) In the classical case (HTop, HPol), for every topological space X one considers the inverse system (its Cechˇ system [13,18]) in HPol, formed by the geometrical realizations of the nerves of numerable coverings of X, directed by refinement, with bonding morphisms the homotopy classes of canonical projections. Then, the homotopy classes of the various canonical maps give an HPol-expansion of X, denoted by X → C(X)ˇ . (2) One more interesting example, without homotopy, is given by the pair of categories (Top, PMet),whereTop is the category of topological spaces and PMet is its subcategory of pseudometrizable spaces. Given a topological space X, one considers the inverse system X = (Xi,xii ,I), where each Xi is a pseudometrizable space having the same underlying set as X, with topology coarser than that of X, while the bonding morphisms xii are all identity maps. The family of the identity maps pi : X → Xi ,definesaPMet-expansion p : X → X. Note that, applying to X above the Hausdorff reflector one gets a metrizable expansion of X, thus obtaining another pre-shape theory (Top, Met),whereMet is the category of metrizable spaces. = = (3) In general, if C Top (respectively C Top 2, the category of Hausdorff spaces), one obtains a pre-shape theory (C, K),forK a full subcategory of C that is closed with respect to finite products and subspaces (respectively closed subspaces) [9,10].

Let (C, K) be a fixed pre-shape theory and let F : K → A, A ∈ Cat∗, be a given functor. If X ∈ C and p : X → X is a K-expansion of X, let us define Fˇ(X) = F∗(X).Insuch a way one obtains a functor F:ˇ C → A. In fact, given f : X → Y in C and q : Y → Y a K-expansion of Y then, by the universal property of p, there exists a unique morphism f : X → Y in Pro K, such that f · p = q · f .ThenFˇ(f ) = F∗(f ). Fˇ is an extension of the given F since, for every X ∈ K, the identity 1X : X → X is a K-expansion, hence Fˇ(X) = F∗(X) = F(X). Fˇ may be called the Cechˇ extension of F, by analogy to what occurs in the topological situation ([4, Chapter 2]; see [13] for an account of Cechˇ extensions of functors defined on some subcategory of HPol).

Proposition 2.3. Let (C, K) be a pre-shape theory. Every functor F:K → A, with A ∈ Cat∗, has a Cechˇ extension F:ˇ C → A. L. Stramaccia / Topology and its Applications 120 (2002) 355–363 359

Example 2.4. (1) Any functor F : Met → A, A ∈ Cat∗, has an extension F:ˇ Top → A. ˇ (2) Let Hn : HPol → Gr be the nth homology group functor. In this case Hn is the nth Cechˇ homology group functor [16]. (3) Let πn : HPol → Gr be the nth homotopy group functor, where Gr is the category ˇ = ∗ → of groups. In such a case πn πn : HTop Gr is the nth shape group functor [16].

Since its beginning, shape came equipped with a close link to the theory of Kan extensions. In fact, Cechˇ cohomology on the category of compact spaces is shape invariant, i.e., it factors through the suitable shape functor and, on the other hand, it is the right Kan extension of simplicial cohomology of compact polyhedra. General results connecting shape to Kan extensions are given in [6,8]. For a very comprehensive treatment of the subject we refer to [4]. In the following we are interested in the case of right Kan extensions along a full embedding, in order to obtain a characterization of pre-shape theories. For a category A ∈ Cat∗ and for a pre-shape theory (C, K), let us denote by AK the category of functors K → A, with morphisms the natural transformations between them. One has a Cechˇ extension functor C:ˇ AK → AC, Cˇ (F) = F,ˇ which is right adjoint to the C K K restriction functor Res : A → A ,Res(S) = S|K. In fact, for an F ∈ A , we have just seen that Res(Cˇ (F)) = F. If S ∈ AC, then there is a natural transformation g :S→ Cˇ (Res S), deﬁned as follows: for every X ∈ C and K-expansion p : X → X, p ={pi : X → Xi | i ∈ ˇ I},letgX :S(X) → C(Res(S))(X), be the unique morphism induced by the morphisms S(pi) :S(X) → S(Xi), i ∈ I. The above data deﬁne a bijection AC S, Cˇ (F) → AK Res(S), F , t → Res(t), which is natural in F ∈ AK and S ∈ AC. Let us recall from [17] that, given a pair (C, K) of categories and a functor F : K → A, the (right) Kan extension of F along the embedding E : K → C is a functor Ran F : C → A, equipped with a natural isomorphism ε :Res(Ran F) → F, in such a way that, for every S ∈ AK,themap AC(S, Ran F) → AK Res(S), F ,σ → ε · Res(σ), is a bijection which is natural in S. The Kan extension is said to be pointwise if, for every X ∈ C, one has δ F Ran F(X) = lim (X ↓ E) −→X K −→ A . A functor T : A → B is said to preserve the right Kan extension Ran F whenever Ran(T · F) = T · (Ran F) holds. It follows that:

Proposition 2.5. Let (C, K) be a pre-shape theory. For every functor F:K → A, A ∈ Cat∗,theCechˇ extension F:ˇ C → A coincides, up to natural isomorphisms, with the 360 L. Stramaccia / Topology and its Applications 120 (2002) 355–363 pointwise right Kan extension Ran F : C → A of F along the embedding E: K ⊂ C. Moreover, Ran F is preserved by every functor G:A → B which commutes with inverse limits.

Proof. The fact that Fˇ = Ran(F) is a right Kan extension follows from the above formula. To see that it is a pointwise extension one should consider the construction of Fasanˇ ˇ inverse limit F(X) = lim(F(Xi ), F(xii ), I), which can be interpreted as the limit of the composition

δ F lim (X ↓ X ) −→X K −→ A , where (X ↓ X ) is the small full subcategory of the comma category (X ↓ K), generated by the K-expansion p : X → X,andδX is the range functor. For the second assertion, let us observe that, since G commutes with inverse limits, then G · lim = lim · Pro G. From this it follows that (G · F)∗ = G · F∗. ✷

Corollary 2.6. Let (C, K) be any pre-shape theory with a proreﬂector P:C → Pro K. Then Ran eK = P and it is preserved by Pro E.

Proof. Let X ∈ C and let p : X → X be a K-expansion of X then, by the previous ∗ proposition, Ran eK(X) = (eK) (X) = lim · ProeK(X) = X = P(X). The last assertion comes from the fact that Pro E commutes with inverse limits [11, 8.6.3]. ✷

Recall that our deﬁnition of Fˇ depends on the choice both of a K-expansion for every X ∈ C and of the way of taking inverse limits in A. Proposition 2.3 tells us that, for every possible choice, we always obtain the same functor F,ˇ up to natural isomorphisms (see [13]).

3. A characterization of pre-shape theories

In this section we shall prove that, whenever a pair (C, K) has a suitable Kan extension property, then it must be a pre-shape theory. This, together with Corollary 2.6, gives a characterization of pre-shape theories. Moreover, it turns out that the Kan extension of a functor F : K → A, A ∈ Cat∗, must coincide with its Cechˇ extension. It follows that the given results also give a formal converse of the result of Deleanu and Hilton (see the Introduction).

Theorem 3.1. Let (C, K) be a pair of categories. Assume that Ran eK exists and is preserved by ProE.Then(C, K) is a pre-shape theory with Ran eK as its proreﬂector.

Proof. In order to prove the assertion we shall prove that the right Kan extensions Ran 1Pro K and Ran Pro E both exist, along the embedding Pro E, and that the relation ProE · Ran 1Pro K = Ran Pro E holds. In fact, in such a case [17, Theorem 2, p. 244], it L. Stramaccia / Topology and its Applications 120 (2002) 355–363 361 follows that Ran 1ProK is a left adjoint to ProE, hence it is a reﬂector [18, 2.4] Pro C → ProK. Finally, Ran 1Pro K · eC will be the required proreﬂector.

1 Pro K ProK ProK Pro E Pro C

Ran 1 Pro E ProK Ran Pro E

Pro C ∗ Let us show that (Ran eK) satisﬁes all the properties which characterize Ran 1Pro K.Given any functor S : ProC → Pro K, we claim that the map ∗ Nat S,(Ran eK) → Nat[S · ProE, 1Pro K], t → t · Pro E, is a bijection which is natural in S. Note that, since Ran eK exists, then, for every H:C → Pro K, there is a bijection

Nat[H, Ran eK]→Nat[H · E, eK], v → v · E, natural in H. Now let τ :S· Pro E → 1Pro K be a natural transformation. It follows that there is a unique natural transformation v : S · eC → Ran eK such that v · E = τ · eK. Finally ∗ ∗ ∗ v ∈ Nat[S,(Ran eK) ],andv · Pro E = τ . ∗ In a similar way one can also show that Ran ProE = (Pro E · Ran eK) . Let us consider the map ∗ Nat T,(Pro E · Ran eK) → Nat[T · ProE, Pro E],σ → σ · Pro E, where T : ProC → Pro C and let σ :T· Pro E → ProE be a given natural transformation. Since Ran(Pro E · eK) exists by assumption then, for every H : C → Pro C,thereisa bijection Nat H, Ran(Pro E · eK) → Nat[H · E, Pro E · eK], u → u · E.

Let H = T · eC : C → ProC and let us note that σ · eK is a natural transformation between T·Pro E·eK = T·eC ·E = H·EandProE·eK. Then, there is a unique natural transformation ∗ ∗ u ∈ Nat[H, Ran(Pro E · eK)] such that u · E = σ · eK. Finally u ∈ Nat[T,(Pro E · Ran eK) ] turns to be the unique natural transformation such that u∗ · Pro E = σ . As for the preservation, one has

∗ (Pro E · Ran eK) = lim · Pro(ProE · Ran eK) = lim · Pro Pro E · Pro Ran eK ∗ = Pro E · lim · ProRan eK = Pro E · (Ran eK) , since Pro E commutes with inverse limits [11, 8.6.3]. It remains to show that Ran 1Pro K · eC = Ran eK is a proreﬂector. Since Ran 1Pro K = ∗ (Ran eK) = lim · ProRan eK and, moreover, ePro K ·Ran eK = Pro Ran eK ·eC, the assertion follows from the fact that lim · ePro K = 1Pro K. ✷ 362 L. Stramaccia / Topology and its Applications 120 (2002) 355–363

Combining Corollary 2.6 with the preceding theorem, we obtain the following characterization of pre-shape theories.

Theorem 3.2. A pair (C, K) is a pre-shape theory if and only if it has the Kan extension property, that is: for every functor F:K → A, A ∈ Cat∗, Ran F exists and it is preserved by every functor G:A → B which commutes with inverse limits.

Proof. As we have already noted, ProK has inverse limits and Pro E commutes with them. ✷

Note that, if a pair (C, K) has the above Kan extension property, then every functor F:K → A, A ∈ Cat∗, actually has a Cechˇ extension.

Example 3.3. Let C ⊂ Top and K be a category of polyhedra contained in C.If(C, K) is a Cechˇ extension pair, as defined in [5], then the homotopy categories of C and K gives rise to a pre-shape theory. The same happens for a Cechˇ extension category (C, K) [5,13], C being a category of Hausdorff paracompact spaces. In fact, in such situations, for every homotopy functor F : K → A, A ∈ Cat∗,theCechˇ and the Kan extensions of F coincide and they are preserved by all functors G : A → B that commute with inverse limits (see the proof of Proposition 2.3). For instance, all of the following pairs of homotopy categories are pre-shape theories: • (HTop, HlfPol), topological spaces, locally finite polyhedra. • (HCompT2, HfPol), compact Hausdorff spaces, finite polyhedra. • (HSMet, HclfPol), separable metric spaces, countable locally finite polyhedra. • (HfdNorm, HfdPol), finite dimensional normal spaces, finite dimensional polyhedra. • (HPar, HfdPol), finite dimensional Hausdorff paracompact spaces, finite dimensional polyhedra.

In the case of a pair (C, K) of topological categories we have mentioned that K is proreﬂective in C whenever it is closed with respect to ﬁnite products and subspaces. The following theorem gives a similar result in the general case.

Theorem 3.4. A pair of categories (C, K) is a pre-shape theory whenever K is closed with respect to limits (possibly large) and Ran eK exists pointwise.

Proof. For every X ∈ C we can write δX eK Pro E(Ran eK)(X) = Pro E lim (X ↓ E) −−−−→ K −−−−→ Pro K δ e Pro E = lim (X ↓ E) −−−−X→ K −−−−K→Pro K −−−−→ Pro C

= Ran(ProE · eK)(X), hence Pro E preserves Ran eK and the assertion follows from Theorem 3.1. ✷ L. Stramaccia / Topology and its Applications 120 (2002) 355–363 363

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